The first eigenvalue of -Laplacian systems with nonlinear boundary conditions

THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS

ISOLATION AND SIMPLICITY FOR THE FIRST EIGENVALUE OF THE p-LAPLACIAN WITH A NONLINEAR BOUNDARY CONDITION

Here Ω is a bounded domain in RN with smooth boundary, ∆pu = div(|∇u|p−2∇u) is the p-Laplacian, and ∂/∂ν is the outer normal derivative. In the linear case, that is for p = 2, this eigenvalue problem is known as the Steklov problem (see [3]). Problems of the form (1.1) appear in a natural way when one considers the Sobolev trace inequality. In fact, the immersionW1,p(Ω) ↪→ Lp(∂Ω) is compact, he...

The ∞−Laplacian first eigenvalue problem

We review some results about the first eigenvalue of the infinity Laplacian operator and its first eigenfunctions in a general norm context. Those results are obtained in collaboration with several authors: V. Ferone, P. Juutinen and B. Kawohl (see [BFK], [BK1], [BJK] and [BK2]). In section 5 we make some remarks on the simplicity of the first eigenvalue of ∆∞: this will be the object of a join...

An Isoperimetric Inequality for the Second Eigenvalue of the Laplacian with Robin Boundary Conditions

We prove that the second eigenvalue of the Laplacian with Robin boundary conditions is minimised amongst all bounded Lipschitz domains of fixed volume by the domain consisting of the disjoint union of two balls of equal volume.

Positive solutions for nonlinear systems of third-order generalized sturm-liouville boundary value problems with $(p_1,p_2,ldots,p_n)$-laplacian

In this work, byemploying the Leggett-Williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-Laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))'  +  a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0  leq t leq 1, alpha_i u...